Just a couple of thoughts about doing this without spending a ton more $$:
I am surprised that the Bauer didn't hold up a bit better than that. Maybe it was just a bad bearing or something, but if it can't take an albie/bonito class fish, it is hard to believe it is going to take a real tuna. Still, if I were on a budget, I would give it a try.
The only, ' Budget,' reel that I would be comfortable with for tuna are the '80's vintage Fin Nors which are readily available on e-bay. # 3's in AR frequently trade in the range of $ 150-300 depending on condition. # 4, 4.5 + 5 are better choices for that application, but scarcer and harder to find. The trick is finding one of the later models which has a palming spool. These can be spotted easily because the spool is drilled with holes, the un-drilled ones do not have palming spools.
If you must use a smaller capacity reel, like your Bauer or the Fin Nor 3, you can fit more backing on it by getting a little creative. On my old 3, I put on a high-density sinking head, and cut an intermediate running line back to 50' long. With a leader, this gives you about 90' of working line and takes up very little space on the reel. With this setup, I was able to easily fit 300 yards of 65lb PowerPro on # 3.
As previously stated, PowerPro is just a brand name of gel-spun braid. For capacity purposes, 65 lb PP is the same diameter as 16lb test monofilament. 50 lb PP is the same as 12 lb. I like the 65 becuse it feels more like real backing in the had than any other braid I have tried, ( But there are plenty I haven't tried.)
For ultimate capacity, Rod Harrison's Bionic Braid is very highly regarded by everyone I know who has used it, but it is pricey.
Lastly I would say that anyone who says you need more than 300 yards of backing on a reel of four-inches or less in diameter, has never tried to retrieve that much line!